Simplicial Structures in Topology

x Preface
same qualitative properties (see [28]). From a more up-to-date point of view, we
may say that topology is the branch of mathematics concerning spaces with a certain
structure (topological spaces), which are invariant under homeomorphisms; in
other words, functions that are injective, surjective, and bicontinuous. The concept
of homeomorphism allows us to group topological spaces into equivalence classes
and consequently, to know in practical terms whether two of them are equal. This is
the main purpose of topology.
This book consists of six chapters. In the first one, we present the basic concepts
in Topology, Group Actions and Category Theory needed for developing the
remainder of the book. The part concerning Category Theory is especially important
since this book has been set in categorial terminology. This chapter could also
serve as a basic text for a mini-course on General Topology.
In the second chapter, we study the category of simplicial complexes Csim, and
two important covariant functors: the geometric realization functor from Csim to
the category of topological spaces, and the homology functor from Csim to the category
of graded Abelian groups. The geometric realization – |K| – of a simplicial
complex K is a polyhedron, assumed to be compact throughout this book. This is
the chapter closest to Poincaré’s initial paper. The Swiss mathematician Leonhard
Euler was among the first ones to study one- and two-dimensional simplicial complexes;
indeed, he used one-dimensional simplicial complexes and their geometric
realization (graphs) in the famous problem about the seven bridges of Königsberg;
later on, he also used two-dimensional simplicial complexes when he noticed that
the relation
v − e + f = 2,
–wherev is the number of vertices, e is the number of edges, and f is the number of
faces – holds for every convex polyhedron in the elementary sense (namely, every
edge is common to two faces and every face leaves the entire polyhedron to one of
its sides).
By defining the so-called Betti Numbers for polyhedra of any dimension, Enrico
Betti gave an initial generalization to the relation above; these are invariant under
homeomorphims and, therefore, useful for classifying polyhedra. Based on these
numbers, Poincaré developed a more complete characterization of polyhedra; in
fact, in his 1895 [27] paper, Poincaré linked the Betti numbers to certain finitely
generated Abelian groups associated with a polyhedron (integral homology groups
of the polyhedron) and pointed out that the Betti numbers are ranks of homology
groups. However, since homology groups are finitely generated Abelian groups,
besides its free part (the one which gives its rank) they also have a torsion part; this
too is an invariant by homeomorphisms. The combination of Betti numbers and
torsion coefficients allows for a more complete analysis of polyhedra.
In Chap. II, homology groups are considered strictly from the simplicial point of
view; in other words, the geometric structure of polyhedra is overlooked. In order
to develop the theory (which is, at this point, of algebraic nature), one needs to
define concepts in Homological Algebra (a branch of Algebra that sprang partly
from Algebraic Topology): among other things, we prove the important Long Exact